CZ:Mathematics Workgroup
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Help plan Mathematics Week!
Work plan white paper
The topics below are part of the division of mathematical knowledge to subdisciplines; they come from the 2000 Math. subject classification of the AMS ^{[1]} and ZMATH ^{[2]}, with some minor edits. Each of these top level topics is a big subject, and it is not necessarily our first priority to write a long article on these entries. Our object with outlining this list is to establish a framework. We recommend that when you write a new article, you should try to find a proper node for it. Do not hesitate to put a link for an article you want to work on soon. Feel free to do the same to request a particular important topic to be covered.
Remarks:
 We kept the original MSC numbering in places.
 No, of course its not the whole MSC tree  not even close to it. We should eventually put as much of it as appropriate.
 In some places we really expect some other workgroups (usually the physicists) to do the work alongside  we state where.
Caveats:
 Do not copy articles from Wikipedia without carefully reading them, verifying both scope and focus. Besides, why not consider writing your article from scratch? Often this permits a better logical structure and more coherence. See CZ:How to convert Wikipedia articles to Citizendium articles
 Keep in mind three audiences when writing an article: general readers, math students and professionals.
Most wanted math entries
From the encyclopedic point of view, the "high priority" articles are probably listed below at the top level of the tree (or in the table of contents). These are, however, relatively wide syntheses. You may want to start with a bit smaller tasks. A collection of important entries to write (of both types) can be found on the Core Articles page (still to be reviewed and reorganized).
Most popular Wikipedia math entries are: Pi, Mathematics , prime number, computer, trapezoid. These come from the list of 1000 most viewed pages in March 2007.^{[3]} This varies heavily over the time.
On Planet Math, the most popular (as of March 2007) were:^{[4]} circle, proof of Markov's inequality, CauchySchwarz inequality, matrix inverse, Banach fixed point theorem, metric space, invariant subspace, function, manifold, eigenvalue, quartic formula, Jensen's inequality, cross product, real number, differential equation, gradient, natural number, Jacobian matrix, GramSchmidt orthogonalization, rational number.
The classification
00XX General
(for calculus see 26XX Real functions below)
 elementary mathematics (preuniversity level)
 Elementary functions
 trigonometric function
 point line plane solid stubs
 elementary algebra
 for suggestions of useful articles to write see also here (precalculus or geometry sections)
01XX History and biography
03XX Mathematical logic and foundations
05XX Combinatorics
06XX Order, lattices, ordered algebraic structures
 06Axx Ordered sets
 06Bxx Lattices
 06Cxx Modular lattices, complemented lattices
 06Dxx Distributive lattices
 06Exx Boolean algebras (Boolean rings)
 06Fxx Ordered structures
08XX General algebraic systems
11XX Number theory
 11Axx Elementary number theory {For analogues in number fields, see 11R04}
 11Bxx Sequences and sets
 11Cxx Polynomials and matrices
 11Dxx Diophantine equations [See also 11Gxx, 14Gxx]
 11Exx Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63}
 11Fxx Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45}
 11Gxx Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14xx, 14Gxx, 14Kxx]
 11Hxx Geometry of numbers {For applications in coding theory, see 94B75}
 11Jxx Diophantine approximation, transcendental number theory [See also 11K60]
 11Kxx Probabilistic theory: distribution modulo 1; metric theory of algorithms
 11Lxx Exponential sums and character sums {For finite fields, see 11Txx}
 11Mxx Zeta and functions: analytic theory
 11Nxx Multiplicative number theory
 11Pxx Additive number theory; partitions
 11Rxx Algebraic number theory: global fields {For complex multiplication, see 11G15}
 11Sxx Algebraic number theory: local and adic fields
 11Txx Finite fields and commutative rings (numbertheoretic aspects)
 11Uxx Connections with logic
 11Yxx Computational number theory [See also 1104]
12XX Field theory and polynomials
 12Dxx Real and complex fields
 12Exx General field theory
 12E05 Polynomials (irreducibility, etc.)
 12E10 Special polynomials
 12E12 Equations
 12E15 Skew fields, division rings
 12E20 Finite fields (fieldtheoretic aspects)
 12E25 Hilbertian fields; Hilbert's irreducibility theorem
 12E30 Field arithmetic
 12Fxx Field extensions
 12Gxx Homological methods (field theory)
 12Hxx Differential and difference algebra
 12Jxx Topological fields
 12Kxx Generalizations of fields
 12Lxx Connections with logic
 12Yxx
 12Y05 Computational aspects of field theory and polynomials
13XX Commutative rings and algebras
14XX Algebraic geometry
 14Axx Foundations
 14A10 Varieties and morphisms
 14A15 Schemes and morphisms
 14A20 Generalizations (algebraic spaces, stacks)
 14A22 Noncommutative algebraic geometry
 14Bxx Local theory
 14B05 Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
 14B07 Deformation of singularities [See also 14D15, 32S30]
 14B10 Infinitesimal methods [See also 13D10]
 14B12 Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10]
 14B15 Local cohomology [See also 13D45, 32C36]
 14B20 Formal neighborhoods
 14B25 Local structure of morphisms: étale morphism, flat morphism, etc. [See also 13B40]
 14Cxx Cycles and subschemes
 14C05 Parametrization (Chow schemes and Hilbert schemes)
 14C15 Chow groups and rings
 14C17 Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
 14C20 Divisors, linear systems, invertible sheaves
 14C21 Pencils, nets, webs [See also 53A60]
 14C22 Picard groups
 14C25 Algebraic cycles
 14C30 Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture
 14C34 Torelli problem [See also 32G20]
 14C35 Applications of methods of algebraic Ktheory [See also 19Exx]
 14C40 RiemannRoch theorems [See also 19E20, 19L10]
 14Dxx Families, fibrations
 14D05 Structure of families (PicardLefschetz, monodromy, etc.)
 14D06 Fibrations, degenerations
 14D07 Variation of Hodge structures [See also 32G20]
 14D10 Arithmetic ground fields (finite, local, global)
 14D15 Formal methods; deformations [See also 13D10, 14B07, 32Gxx]
 14D20 Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
 14D22 Fine moduli spaces and coarse moduli spaces
 14Exx Birational geometry
 14E05 Rational and birational maps
 14E07 Birational automorphisms, Cremona group and generalizations
 14E08 Rationality questions
 14E15 Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
 14E20 Coverings [See also 14H30]
 14E22 Ramification problems [See also 11S15]
 14E25 Embeddings
 14E30 Minimal model program (Mori theory, extremal rays)
 14Fxx (Co)homology theory [See also 13Dxx]
 14F05 Vector bundles, sheaves, related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
 14F10 sheaf of Differentials and other special sheaves [See also 13Nxx, 32C38]
 14F17 Vanishing theorems [See also 32L20]
 14F20 Étale topology Etale cohomology and other Grothendieck topologies and Grothendieck cohomologies
 14F22 Brauer groups of schemes [See also 12G05, 16K50]
 14F25 Classical real and complex cohomology
 14F30 padic cohomology, crystalline cohomology
 14F35 Homotopy theory; fundamental groups [See also 14H30]
 14F40 de Rham cohomology [See also 14C30, 32C35, 32L10]
 14F42 Motivic cohomology
 14F43 Other algebrogeometric (co)homologies (e.g., intersection cohomology, equivariant cohomology, Lawson, Deligne (co)homologies)
 14F45 Topological properties
 14Gxx Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx]
 14Hxx Curves
 14H05 Algebraic functions; function fields [See also 11R58]
 14H10,14H15 moduli [See also 30F10, 32Gxx]
 14H20 Singularities, local rings [See also 13Hxx, 14B05]
 14H25 Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
 14H30 Coverings, fundamental group [See also 14E20, 14F35]
 14H37 Automorphisms
 14H40 Jacobians, Prym varieties [See also 32G20]
 14H42 Theta functions; Schottky problem [See also 14K25, 32G20]
 14H45 Special curves and curves of low genus
 14H50 Plane and space curves
 14H51 Special divisors (gonality, BrillNoether theory)
 14H52 Elliptic curves [See also 11G05, 11G07, 14Kxx]
 14H55 Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]
 14H60 Vector bundles on curves and their moduli [See also 14D20, 14F05]
 14Jxx Surfaces and higherdimensional varieties {For analytic theory, see 32Jxx}
 14J10 Families, moduli, classification: algebraic theory
 14J15 Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
 14J17 Singularities of surfaces [See also 14B05, 14E15]
 14J20 Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
 14J25 Special surfaces {For Hilbert modular surfaces, see 14G35}
 14J26 Rational surfaces and ruled surfaces
 14J27 Elliptic surfaces
 14J28 K3 surfaces and Enriques surfaces
 14J29 Surfaces of general type
 14J30 3folds
 14J32 CalabiYau manifolds, mirror symmetry
 14J35 4folds
 14J40 nfolds (n > 4)
 14J45 Fano varieties
 14J50 Automorphisms of surfaces and higherdimensional varieties
 14J60 Vector bundles on surfaces and higherdimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
 14J70 Hypersurfaces
 14J80 Topology of surfaces (Donaldson polynomials, SeibergWitten invariants)
 14Kxx Abelian varieties and schemes
 14Lxx Algebraic groups {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45}
 14Mxx Special varieties
 14Nxx Projective and enumerative geometry [See also 51xx]
 14Pxx Real algebraic and real analytic geometry
 14Qxx Computational algebraic geometry [See also 12Y05, 13Pxx, 68W30]
 14Rxx Affine geometry
15XX Linear and multilinear algebra; matrix theory
 15A03 Vector spaces, linear dependence, rank
 15A04 Linear transformations, semilinear transformations
 15A06 Linear equations
 15A09 Matrix inversion, generalized inverses
 15A12 Conditioning of matrices
 15A15 Determinants, permanents, other special matrix functions
 15A18 Eigenvalues, singular values, and eigenvectors
 15A21 Canonical forms, reductions, classification
 15A22 Matrix pencils
 15A23 Factorization of matrices
 15A24 Matrix equations and identities
 15A27 Commutativity
 15A29 Inverse problems
 15A30 Algebraic systems of matrices
 15A33 Matrices over special rings (quaternions, finite fields, etc.)
 15A36 Matrices of integers
 15A39 Linear inequalities
 15A42 Inequalities involving eigenvalues and eigenvectors
 15A45 Miscellaneous inequalities involving matrices
 15A48 Positive matrices and their generalizations; cones of matrices
 15A51 Stochastic matrices
 15A52 Random matrices
 15A54 Matrices over function rings in one or more variables
 15A57 Other types of matrices (Hermitian, skewHermitian, etc.)
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
 15A63 Quadratic and bilinear forms, inner products [See mainly 11Exx]
 15A66 Clifford algebras, spinors
 15A69 Multilinear algebra, tensor products
 15A72 Vector and tensor algebra, theory of invariants
 15A75 Exterior algebra, Grassmann algebras
 15A78 Other algebras built from modules
 15A90 Applications of matrix theory to physics
16XX Associative rings and algebras
17XX Nonassociative rings and algebras
18XX Category theory; homological algebra
 18Axx General theory of categories and functors
 18Bxx Special categories
 18Cxx Categories and theories
 18Dxx Categories with structure
 18Exx Abelian categories
 18Fxx Categories and geometry
 18Gxx Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx]
 18G05 Projective objects and injective objects [See also 13C10, 13C11, 16D40, 16D50]
 18G10 Resolutions; derived functors [See also 13D02, 16E05, 18E25]
 18G15 Ext and Tor, generalizations, Künneth formula [See also 55U25]
 18G20 Homological dimension [See also 13D05, 16E10]
 18G25 Relative homological algebra, projective classes
 18G30 Simplicial sets, simplicial objects (in a category) [See also 55U10]
 18G35 Chain complexes [See also 18E30, 55U15]
 18G40 Spectral sequences, hypercohomology [See also 55Txx]
 18G50 Nonabelian homological algebra
 18G55 Homotopical algebra
 18G60 Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]*20XX Group theory and generalizations
19XX Ktheory
20XX Group theory and generalizations
 20Axx Foundations
 20Bxx Permutation groups
 20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)]
 20Dxx Abstract finite groups
 20Exx Structure and classification of infinite or finite groups
 20Fxx Special aspects of infinite or finite groups
 20Gxx Linear algebraic groups (classical groups) {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55}
 20Hxx Other groups of matrices [See also 15A30]
 20Jxx Connections with homological algebra and category theory
 20Kxx Abelian groups
 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) {For sets with a single binary operation, see 20N02; for topological groupoids, see 22A22, 58H05}
 20Mxx Semigroups
 20Nxx Other generalizations of groups
 20P05 Probabilistic methods in group theory [See also 60Bxx]*22XX Topological groups, Lie groups
22XX Topological groups, Lie groups
For some useful suggestions see: Lie groups topics
26XX Real functions
 Mean value theorem
 for some more suggestions of useful articles to write see also list of mathematical topics
28XX Measure and integration
 28Axx Classical measure theory
 28A05 Classes of sets (Borel fields, $\sigma$rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
 28A10 Real or complexvalued set functions
 28A12 Contents, measures, outer measures, capacities
 28A15 Abstract differentiation theory, differentiation of set functions [See also 26A24]
 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
 28A25 Integration with respect to measures and other set functions
 28A33 Spaces of measures, convergence of measures [See also 46E27, 60Bxx]
 28A35 Measures and integrals in product spaces
 28A50 Integration and disintegration of measures
 28A51 Lifting theory [See also 46G15]
 28A60 Measures on Boolean rings, measure algebras [See also 54H10]
 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
 28A78 Hausdorff and packing measures
 28A80 Fractals [See also 37Fxx]
 28A99 None of the above, but in this section
 28Bxx Set functions, measures and integrals with values in abstract spaces
 28Exx Miscellaneous topics in measure theory
30XX Functions of a complex variable
31XX Potential theory
 31Axx Twodimensional theory
 31A05 Harmonic, subharmonic, superharmonic functions
 31A10 Integral representations, integral operators, integral equations methods
 31A15 Potentials and capacity, harmonic measure, extremal length [See also 30C85]
 31A20 Boundary behavior (theorems of Fatou type, etc.)
 31A25 Boundary value and inverse problems
 31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation
 31A35 Connections with differential equations
 31A99 None of the above, but in this section
 31Bxx Higherdimensional theory
 31B05 Harmonic, subharmonic, superharmonic functions
 31B10 Integral representations, integral operators, integral equations methods
 31B15 Potentials and capacities, extremal length
 31B20 Boundary value and inverse problems
 31B25 Boundary behavior
 31B30 Biharmonic and polyharmonic equations and functions
 31B35 Connections with differential equations
 31B99 None of the above, but in this section
 31Cxx Other generalizations
 31D05 Axiomatic potential theory
32XX Several complex variables and analytic spaces
33XX Special functions
(33XX deals with the properties of functions as functions) {For orthogonal functions, see 42Cxx; for aspects of combinatorics see 05Axx; for numbertheoretic aspects see 11XX; for representation theory see 22Exx}
34XX Ordinary differential equations
 see the current article on differential equations
35XX Partial differential equations
37XX Dynamical systems and ergodic theory
39XX Difference and functional equations
40XX Sequences, series, summability
 40Axx Convergence and divergence of infinite limiting processes
 40B05 Multiple sequences and series {(should also be assigned at least one other classification number in this section)}
 40Cxx General summability methods
 40Dxx Direct theorems on summability
 40Exx Inversion theorems
 40F05 Absolute and strong summability
 40Gxx Special methods of summability
 40H05 Functional analytic methods in summability
 40J05 Summability in abstract structures [See also 43A55, 46A35, 46B15]
41XX Approximations and expansions
42XX Fourier analysis
 4204 Explicit machine computation and programs (not the theory of computation or programming)
 42Axx Fourier analysis in one variable
 42Bxx Fourier analysis in several variables {For automorphic theory, see mainly 11F30}
 42Cxx Nontrigonometric Fourier analysis
43XX Abstract harmonic analysis
44XX Integral transforms, operational calculus
45XX Integral equations
46XX Functional analysis
47XX Operator theory
49XX Calculus of variations and optimal control; optimization
51XX Geometry
 for a list of possible suggestions see list of geometry topics
52XX Convex and discrete geometry
53XX Differential geometry
54XX General topology
55XX Algebraic topology
57XX Manifolds and cell complexes
58XX Global analysis, analysis on manifolds
60XX Probability theory and stochastic processes
 60Axx Foundations of probability theory
 60Bxx Probability theory on algebraic and topological structures
 60C05 Combinatorial probability
 60D05 Geometric probability, stochastic geometry, random sets [See also 52A22, 53C65]
 60Exx Distribution theory [See also 62Exx, 62Hxx]
 60Fxx Limit theorems [See also 28Dxx, 60B12]
 60Gxx Stochastic processes
 60Hxx Stochastic analysis [See also 58J65]
 60Jxx Markov processes
 60Kxx Special processes
62XX Statistics
65XX Numerical analysis
68XX Computer science
(do we leave it for the computers Workgorup ?)
70XX Mechanics of particles and systems
(do we leave it for physics Workgroup??)
74XX Mechanics of deformable solids
(do we leave it for physics Workgroup??)
76XX Fluid mechanics
(do we leave it for physics Workgroup??)
78XX Optics, electromagnetic theory
{For quantum optics, see 81V80} (do we leave it for physics Workgroup??)
80XX Classical thermodynamics, heat transfer
(do we leave it for physics Workgroup??)
81XX Quantum theory
82XX Statistical mechanics, structure of matter
(do we leave it for physics Workgroup??)
83XX Relativity and gravitational theory
85XX Astronomy and astrophysics
(do we leave it for physics Workgroup??)
86XX Geophysics
(do we leave it for physics Workgroup??)
90XX Operations research, mathematical programming
91XX Game theory, economics, social and behavioral sciences
(do we leave it for economy Workgroup??)
92XX Biology and other natural sciences
(do we leave it for biology Workgroup??)